Question

# A class has three teachers, Mr. $$P$$, Ms. $$Q$$ and $$Mrs. R$$ and six students $$A, B, C, D, E, F$$. Number of ways in which they can be seated in a line of $$9$$ chairs, if between any two teachers there are exactly two students, is $$k!(18)$$, then the value of $$k$$ is.

Solution

## Let $$T$$ and $$S$$ denotes teacher and student respectively.Then we have following possible patterns according to question(i) $$T\ S\ S\ T\ S\ S\ T\ S\ S$$(ii) $$S\ T\ S\ S\ T\ S\ S\ T\ S$$(iii) $$S\ S\ T\ S\ S\ T\ S\ S\ T$$Hence total number of arrangements are $$3 \cdot (3!) 6! = 18\times 6! \Rightarrow k = 6$$.Mathematics

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